Summary
Highlights
The video extends previous concepts of probability mass function (FX) and cumulative distribution function (CDF) for discrete and continuous random variables to bivariate cases, which involve two random variables defined together on a joint sample space. The discussion will start with discrete cases due to their simpler nature.
For two discrete random variables, X and Y, the probability distribution is denoted as P(X=x, Y=y), representing the probability of the intersection of events X and Y occurring simultaneously.
An example is introduced: a bottle contains 3 aspirin, 2 sedative, and 4 laxative caplets. Two caplets are selected at random. X represents the number of aspirin, and Y represents the number of sedatives. The goal is to find probabilities for all possible pairs of X and Y.
A tree diagram is used to visualize the selection process. The first draw can be aspirin (3/9), sedative (2/9), or laxative (4/9). Since the caplet is kept, the total number for the second draw reduces to 8, leading to conditional probabilities for the second draw based on the first.
All possible outcomes from the tree diagram (e.g., Aspirin-Aspirin, Aspirin-Sedative) are listed in the sample space. These outcomes are then mapped to the defined random variables (x, y), where x is the number of aspirins and y is the number of sedatives. For example, (Aspirin, Aspirin) maps to (2, 0).
A two-dimensional table is created with possible values for X (0, 1, 2) and Y (0, 1, 2). The probabilities for each (x, y) combination are calculated by multiplying the probabilities along the corresponding paths in the tree diagram and summing probabilities for identical (x, y) pairs. Some combinations have a probability of 0, as they are impossible with only two draws.
The video concludes by formally defining the joint probability distribution (or joint probability mass function) for discrete random variables as f(x, y) = P(X=x, Y=y). It emphasizes two conditions: all probabilities must be non-negative, and the sum of all probabilities must equal 1.